Area of an isosceles triangle

Pause this video and see if you can find the area of this triangle. I'll give you two hints: recognize this is an isosceles triangle, and another hint is that the Pythagorean theorem might be useful.

All right, now let's work through this together. So we might all remember that the area of a triangle is equal to one-half times our base times our height. They give us our base. Our base right over here is 10. But what is our height? Our height would be, let me do this in another color, our height would be the length of this line right over here.

So if we can figure that out, then we can calculate what one-half times the base, 10, times the height is. But how do we figure out this height? Well, this is where it's useful to recognize that this is an isosceles triangle. An isosceles triangle has two sides that are the same, and so these base angles are also going to be congruent.

If we drop an altitude right over here, which is the whole point—that's the height—we know that these are going to be right angles. If we have two triangles where two of the angles are the same, we know that the third angle is going to be the same. So that is going to be congruent to that.

If you have two triangles, and this might be obvious already to you intuitively, where look, I have two angles in common and the side in between them is common; it's the same length, well that means that these are going to be congruent triangles.

Now what's useful about that is if we recognize that these are congruent triangles, notice they both have a side 13. They both have a side whatever this length of blue is. Then they're both going to have a side length that's half of this 10. So this is going to be 5, and this is going to be 5.

How was I able to deduce that? You might just say, "Oh, that feels intuitively right." I was a little bit more rigorous here, where I said, "Oh, if these are two congruent triangles, then we're going to split this 10 in half because this is going to be equal to that, and they add up to 10."

All right, now we can use the Pythagorean theorem to figure out the length of this blue side, or the height. If we call this h, the Pythagorean theorem tells us that h squared plus 5 squared is equal to 13 squared.

h squared plus 5 squared plus 5 squared is going to be equal to 13 squared, which is going to be equal to our longest side, our hypotenuse squared. So let's see, 5 squared is 25, 13 squared is 169. We can subtract 25 from both sides to isolate h squared.

So let's do that, and what are we left with? We are left with h squared equal to these cancelled out. 169 minus 25 is 144. Now if you're doing it purely mathematically, h could be plus or minus 12, but we're dealing with the distance, so we'll focus on the positive.

So h is going to be equal to the principal root of 144. So h is equal to 12. Now we aren't done. Remember, they don't want us to just figure out the height here; they want us to figure out the area. Area is one-half base times height.

Well, we already figured out that our base is this 10 right over here. Let me do this in another color. So our base is that distance which is 10, and now we know our height. Our height is 12.

So now we just have to compute one-half times 10 times 12. Well that's just going to be equal to one-half times 10 is 5 times 12 is 60. 60 square units, whatever our units happen to be, that is our area.